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Openfoam constraints and assumptions

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5.1 Stress analysis of a plate with a hole


It is loaded with a uniform traction of 10 over its left and right faces as shown in Figure 5. Two symmetry planes can be identified for this geometry and therefore the solution domain need only cover a quarter of the geometry, shown by the shaded area in Figure 5.

Figure 5. In a Cartesian coordinate system there are two possible assumptions to take in regard to the behaviour of the structure in the third dimension: 1 the plane stress condition, in which the stress components acting out of the 2D plane are assumed to be negligible; 2 the plane strain condition, in which the strain components out of the 2D plane are assumed negligible.

The plane stress condition is appropriate for solids whose third dimension is thin as in this case; the plane strain condition is applicable for solids where the third dimension is thick.

An analytical solution exists for loading of an infinitely large, thin plate with a circular hole. The solution for the stress normal to the vertical plane of symmetry is 5. At the end of the tutorial, the user can: investigate the sensitivity of the solution to mesh resolution and mesh grading; and, increase the size of the plate in comparison to the hole to try to estimate the error in comparing the analytical solution for an infinite plate to the solution of this problem of a finite plate.

The block structure for the part of the mesh in the plane is shown in Figure 5. As already mentioned in section 2. Therefore a dimension of the block in the direction has to be chosen; here, 0. It does not affect the solution since the traction boundary condition is specified as a stress rather than a force, thereby making the solution independent of the cross-sectional area.

These are specified under the edges keyword entry which is a list of non-straight edges. The syntax of each list entry begins with the type of curve, including arc , simpleSpline , polyLine etc.

In this example, all the edges are circular and so can be specified by the arc keyword entry. The following entries are the labels of the start and end vertices of the arc and a point vector through which the circular arc passes.

The blocks in this blockMeshDict do not all have the same orientation. As can be seen in Figure 5. This means care must be taken when defining the number and distribution of cells in each block so that the cells match up at the block faces.

The left and down patches are both a symmetry plane. Since this is a geometric constraint, it is included in the definition of the mesh , rather than being purely a specification on the boundary condition of the fields. Therefore they are defined as such using a special symmetryPlane type as shown in the blockMeshDict. The frontAndBack patch represents the plane which is ignored in a 2D case.

Again this is a geometric constraint so is defined within the mesh, using the empty type as shown in the blockMeshDict. For further details of boundary types and geometric constraints, the user should refer to User Guide section?? The remaining patches are of the regular patch type. The mesh should be generated using blockMesh and can be viewed in paraFoam as described in section 2. It should appear as in Figure 5.

For a stress analysis case without thermal stresses, only displacement D needs to be set. Similarly the frontAndBack patch is declared empty. The other patches are traction boundary conditions, set by a specialist traction boundary type. The traction boundary conditions are specified by a linear combination of: 1 a boundary traction vector under keyword traction ; 2 a pressure that produces a traction normal to the boundary surface that is defined as negative when pointing out of the surface, under keyword pressure.

The up and hole patches are zero traction so the boundary traction and pressure are set to zero. For the right patch the traction should be and the pressure should be 0. For this problem, we need to specify the mechanical properties of steel given in Table 5. In the mechanical properties dictionary, the user must also set planeStress to yes.

The user specifies at run time whether OpenFOAM should solve the thermal equation by the thermalStress switch in the thermalProperties dictionary. This dictionary also sets the thermal properties for the case, e. Therefore we must set the thermalStress keyword entry to no in the thermalProperties dictionary. For this case, the startTime is 0. The time step is not important since this is a steady state case; in this situation it is best to set the time step deltaT to 1 so it simply acts as an iteration counter for the steady-state case.

The endTime , set to , then acts as a limit on the number of iterations. The writeInterval can be set to. Firstly, the problem we are analysing is steady-state so the user should select SteadyState for the time derivatives in timeScheme. This essentially switches off the time derivative terms. Not all solvers, especially in fluid dynamics, work for both steady-state and transient problems but solidDisplacementFoam does work, since the base algorithm is the same for both types of simulation.

The momentum equation in linear-elastic stress analysis includes several explicit terms containing the gradient of displacement. The calculations benefit from accurate and smooth evaluation of the gradient. The user should first look at the solvers sub-dictionary and notice that the choice of solver for D is GAMG.

The solver tolerance should be set to for this problem. The solver relative tolerance, denoted by relTol , sets the required reduction in the residuals within each iteration.

It is uneconomical to set a tight low relative tolerance within each iteration since a lot of terms in each equation are explicit and are updated as part of the segregated iterative procedure. Therefore a reasonable value for the relative tolerance is , or possibly even higher, say , or in some cases even as in this case.

Firstly there is nCorrectors which specifies the number of outer loops around the complete system of equations, including traction boundary conditions within each time step. We can therefore set nCorrectors to 1. The D keyword specifies a convergence tolerance for the outer iteration loop, i. It should be set to the desired solver tolerance specified earlier, for this problem. The final residual should always be less than 0. Once both initial residuals have dropped below the convergence tolerance of the run has converged and can be stopped by killing the batch job.

The solidDisplacementFoam solver outputs the stress field as a symmetric tensor field sigma. This is consistent with the way variables are usually represented in OpenFOAM solvers by the mathematical symbol by which they are represented; in the case of Greek symbols, the variable is named phonetically.

For post-processing individual scalar field components, , etc. The stresses can be viewed in paraFoam as shown in Figure 5. We would like to compare the analytical solution of Equation 5. We therefore must output a set of data of along the left edge symmetry plane of our domain. The user may generate the required graph data using the postProcess utility, using a sets function object. The utility can be driven from a user-supplied file located in the system directory, whose entries are summarised in User Guide Table??

The data is written into files within time subdirectories of a sets directory, e. In an application such as GnuPlot , one could type the following at the command prompt would be sufficient to plot both the numerical data and analytical solution: plot [0. Use mapFields to map the final coarse mesh results from section 5. Design the mesh so that the ratio of sizes between adjacent cells is no more than 1. Mesh grading is described in section 2. Again use mapFields to map the final coarse mesh results from section 5.

Compare the results with those from the analytical solution and previous calculations. Can this solution be improved upon using the same number of cells with a different solution? Therefore this solution is not completely accurate for a finite sized plate. To estimate the error, increase the plate size while maintaining the hole size at the same value.

Home Documentation Tutorial Guide 5 Stress analysis. Tutorial Guide. Tutorial Guide Contents 1 Introduction 1. Table 5.


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openfoam constraints and assumptions

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It is particularly useful for rotating geometries. In previous versions of OpenFOAM, there was a requirement that the patches conform fairly closely to one another.

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Scalability of OpenFOAM Density-Based Solver with Runge-Kutta Temporal Discretization Scheme.

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